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Theorem cocanfo 36492
Description: Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
cocanfo (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐺 = 𝐻)

Proof of Theorem cocanfo
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 768 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → (𝐺𝐹) = (𝐻𝐹))
21fveq1d 6883 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → ((𝐺𝐹)‘𝑦) = ((𝐻𝐹)‘𝑦))
3 simpl1 1192 . . . . . . 7 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐹:𝐴onto𝐵)
4 fof 6795 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
53, 4syl 17 . . . . . 6 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐹:𝐴𝐵)
6 fvco3 6979 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐺𝐹)‘𝑦) = (𝐺‘(𝐹𝑦)))
75, 6sylan 581 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → ((𝐺𝐹)‘𝑦) = (𝐺‘(𝐹𝑦)))
8 fvco3 6979 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
95, 8sylan 581 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
102, 7, 93eqtr3d 2781 . . . 4 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)))
1110ralrimiva 3147 . . 3 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → ∀𝑦𝐴 (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)))
12 fveq2 6881 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐺‘(𝐹𝑦)) = (𝐺𝑥))
13 fveq2 6881 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐻‘(𝐹𝑦)) = (𝐻𝑥))
1412, 13eqeq12d 2749 . . . . 5 ((𝐹𝑦) = 𝑥 → ((𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)) ↔ (𝐺𝑥) = (𝐻𝑥)))
1514cbvfo 7274 . . . 4 (𝐹:𝐴onto𝐵 → (∀𝑦𝐴 (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
163, 15syl 17 . . 3 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → (∀𝑦𝐴 (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
1711, 16mpbid 231 . 2 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥))
18 eqfnfv 7021 . . . 4 ((𝐺 Fn 𝐵𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
19183adant1 1131 . . 3 ((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
2019adantr 482 . 2 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → (𝐺 = 𝐻 ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
2117, 20mpbird 257 1 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐺 = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  ccom 5676   Fn wfn 6530  wf 6531  ontowfo 6533  cfv 6535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-fo 6541  df-fv 6543
This theorem is referenced by: (None)
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